It may be beneficialhelpful to compare this law to another similrasimilar law concerning sacrifices,
Do the same with your cattle and your sheep. Let them stay with their mothers for seven days, but give them to me on the eighth day.
Some have regarded this seven-day period as a humane treatment of the animals, but i don't see it as such. The motive for the seven-day period is most probably a ritual concern than it is a purely ethical one; the ancients may have regarded young animals as ritually unfit for sacrifice similar to a blemished animal (perhaps it was considered too weak and unstable and incapable of surviving on its own). The number seven signified completion in the ancient world, so once the animal went through the seven-day period it was considered stable, healthy and fit for divine consumption.
Now you may ask, "what has this all got to do with circumcision?"
There is much evidence that circumcision in ancient Canaan was considered a kind of sacrifice to the gods, and it is reasonable to assume that the Israelite'sIsraelites too regarded it as such. In fact there is evidence in the bible itself that circumcision was regarded as a kind of sacrifice. See Exodus 4:24-27 and also here for more on circumcision and its sacrificial nature. Hence the need to wait seven days before the child is fit for the circumcision ritual.